On the Parametric Solutions of the Diophantine Equation x³+1=Dy²

By use of the parameter method,x ³+1=Dy ²(D>0)is decomposed into an equation set composed of a onedegree equation and a quadratic equation, whose solutions are expressed by a parameter, and by giving this parameter a value, the nontrivial solutions of this indeterminate equation are thus obtained. This article first discusses the ways for determining the solutions of x ³+1=Dy ²(D>0)when D is Irreducible. Then the correspondent nontrivial solutions of x ³+1=Dy ²(D>0)are presented when the value of D is different. Finally ways for determining the non-trivial solutions for x ³+1=Dy ²(D>0)are examined when D is reducible, and the non-trivial solutions for the correspondent indeterminate equations are also put forth.